Problem: What is the smallest possible value of the sum $\lvert x + 2\rvert + \lvert x + 4\rvert + \lvert x + 5\rvert$?
Answer: For $x \le -5,$
\[|x + 2| + |x + 4| + |x + 5| = -(x + 2) - (x + 4) - (x + 5) = -3x - 11.\]For $-5 \le x \le -4,$
\[|x + 2| + |x + 4| + |x + 5| = -(x + 2) - (x + 4) + (x + 5) = -x - 1.\]For $-4 \le x \le -2,$
\[|x + 2| + |x + 4| + |x + 5| = -(x + 2) + (x + 4) + (x + 5) = x + 7.\]For $x \ge -2,$
\[|x + 2| + |x + 4| + |x + 5| = (x + 2) + (x + 4) + (x + 5) = 3x + 11.\]Thus, the function is decreasing on $(-\infty,4]$ and increasing on $[4,\infty),$ so the minimum value occurs at $x = -4,$ which is $\boxed{3}.$